Assignment 1: Logarithm at the End of Multiplication

by R. Adam Molnar

This exploration could be completed by students in Algebra during a unit on logarithms. However, the motivation comes from multiple regression analysis in statistics. The natural log transformation is the most common transformation used in multiple regression. The main purpose of this write-up is to visually illustrate some properties of logarithms.

Motivation

In Algebra, the logarithm is typically introducted as the anti-exponential. It's related to place value, power, and scientific notation. There are also some applications, including compound interest. In regression analysis it has a very important role. This arises from two properties of any logarithm. For any positive values a and b, the logarithm converts multiplication to addition:

and for any positive a and any value (positive, zero, or negative) b, the logarithm converts powers to multiplication:

Graphing the Logarithm

Let's look at the default mathematical logarithm, found by graphing y = \log(x) in Graphing Calculator. It flattens out fairly quickly as x increases.

There are different types of logarithms, because the value depends on the base. In mathematical language, the base is b in b^{\log x} = x. For example, since 10^{2} = 100, the logarithm of 100 base 10 = 2. But the logarithm of 100 base 4 is roughly 3.32, since 4^{3.32} \approx 100. If you have a logarithm in a given base b, we can translate it to base a by dividing by the logarithm of new base a in base b. The subscript on a log indicates the base.

Another difficulty is that the default logarithm changes from field to field. Mathematicians usually use base 10 logarithms, like our number system. Statisticians generally refer to the natural logarithm, base e, because a calculus fact tells us the derivative of e^{x} remains e^{x}. Computer Scientists most frequently use the logarithm base 2 because that counts binary bits. This can make things a little confusing. Here's a graph of all three bases. Mathematical base 10 is colored purple on the bottom; statistical base e is black in the middle; and computer science base 2 is light blue on the top.

For the rest of this page, we're going to consider the statistical logarithm, base e. As a reminder, one way to define e is as a limit: e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n} \right)^{n}. If you don't know limits, just think of e as a special mathematical number developed from a calculus fact. It's an irrational number, roughly equivalent to 2.718.

Multipication to Addition

First, let's look at multiplication with a specific example. The graph below displays y = \ln(x) in purple and y = \ln(2x) in blue. The blue curve is higher, but it isn't twice as high.

The natural log of 20 is very close to 3 (about 2.9957); there are lines that mark that point on the graph. Moving upwards from the purple graph to the blue graph, we see that y = \ln(2 \bullet 20) is a little under 3.75. That line crosses the purple curve at 40, as it should. Furthermore, there's a line at the natural log of 2, about 0.693. The distance from the x-axis to the purple graph at 2, y = \ln(2), equals the distance from the purple graph at 20 to the blue graph at 20. Visually, as the arrows indicate, y = \ln(2 \bullet 20) - \ln(20) = \ln(2).

Shown below is a graph with various multiples of constant B inside y = \ln(B x). From top to bottom, as labeled, the values of B are 5, 2, 1, 0.5, and 0.2. The difference is mostly translation; there's not change in the shape of the curve.

If you want to experiment with the effect, try the graphing calculator file, if you have the free Graphing Calculator Viewer for Macintosh or PC.

Exponent to Multiplication

The other important relationship of the logarithm is that an exponent becomes multiplication. Let's look at an example, where we compare the black line \ln(x) to the purple line \ln(x^2) . As shown with the grey lines, the distance from zero to \ln(4) is the same as from \ln(4) to \ln(4^2) = \ln(16) .

Shown below is a graph with various exponents of B as part of y = \ln(x^B) = B \ln (x) . From top to bottom, the values of B are 5, 2, 1, 0.5, and 0.2; the labels show the exponent brought down into the front. Compared to the graph of y = \ln (Bx) , there's a much greater variation in shape.

One important difference between multiplication and exponent is that exponents can be negative, unlike multiplication. The logarithm of a negative value is not a real number, so if x > 0, y = \ln (-2 x) does not exist. However, x > 0, y = \ln (x^{-2}) = -2 \ln (x) does exist. The graph for the negative power is the reflection across the X-axis of the graph for the positive power.

If you want to watch or download a Quicktime animation of the effect for various values of the exponent, from -4 to +4, click on lnpower.mov.

References and Notes

For logarithms in statistical data transformation, Wikipedia is fairly good. While non-math pages on Wikipedia are of uneven quality, in general Wiki math has high standards. For logarithms in computer science, try the binary logarithm page. Graphing Calculator only provides functions to graph log base 10 and log base e, so I used the transformation to get the computer science log base 2.

The title was adapted from Masashi Harada's experimental electronic album Obliteration at the End of Multiplication.